Campos–de Mello total coloring conjecture for powers of cycles

Let $C_n^k$ be the $k$th power of the cycle on $n$ vertices, with $2\leq k<\lfloor n/2\rfloor$. Its maximum degree is denoted by $\Delta(C_n^k)$, and $\chi”(C_n^k)$ is its total chromatic number.

Campos–de Mello's conjecture. If $G=C_n^k$, then

$$\chi”(G)=\begin{cases} \Delta(G)+2, & \text{if } k>\frac{n}{3}-1 \text{ and } n \text{ is odd},\\ \Delta(G)+1, & \text{otherwise}. \end{cases}$$

The source states that the total coloring conjecture has been verified for some even-order powers of cycles and that these graphs admit polynomial-time total colorings, while the displayed classification remains presented as a conjecture. The source gives no resolution status for this classification.